Just focus on passing probation. Everything else can wait, but I do think you should talk to your GP about some counselling.
If you don't have any already, make sure you're saving for at least 2 months' rent. This will help you rent.
It might be a scary thought, especially if you don't the emotional support you should from your parents, but I think you're going to have to rent with strangers.
Try to get a female-only house. I've lived with men, and although there were some nice ones I'd live with again, I wouldn't do that again if I houseshared. I came across far too many lazy, sexist mummy's boys who thought that because they had a female housemate, cleaning of any kind (including changing the bins, ever) was beneath them.
Make sure that any house you see has a lock on the bedroom (might be important for insurance purposes. Try to get a house where you all pay rent individually to the landlords, all with separate contracts. Try to avoid a situation whereby you all pay as one entity, because if one person can't pay, it'll become a big problem, rather it being the landlord's problem. If you can get a house that's all inclusive including bills (which is rare), even better if you can afford it. It'll make your life simpler whilst you get used to renting & living on your own.
Go see at least 5 houses. It's important to have that variety as I saw some places where I knew I'd feel lonely (despite the nice people there), and the house I settled in just felt right straight away. Be prepared for it to feel like an interview, because in many ways it is. It's hard to not to feel rejected if people don't want to rent with you, but keep trying. Try not to arrive deflated or sad looking. People will be looking for a relaxed but positive energy when they talk to you, as well as similar lifestyles.
I'm afraid you'll need to leave your dog at home, if they're willing to have it. Lots of people your age have to do this when they go to university - they go to uni, leave home, and their family pet stays with the family for good.
I know I should really go no contact with my parents, or my Dad at least
Who on earth has given you the impression that that you should do this??
For some people, yes, it's the best thing for them, or they almost don't have the option as their parents are violent or as seriously damaging in their lives. If you're not at that point yet, you don't have to go no contact. It doesn't have to be that extreme. You don't even know what it'll be like living on your own yet!!
Your parents, even if toxic, are an important practical support, even if they're rubbish or can be abusive emotionally. Don’t cut ties if you don't have to, especially if you don't have a strong support network.